36 lines
945 B
Text
36 lines
945 B
Text
/-
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Copyright (c) 2021 Microsoft Corporation. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Leonardo de Moura
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-/
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prelude
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import Init.Data.Nat.Div
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namespace Nat
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private def gcdF (x : Nat) : (∀ x₁, x₁ < x → Nat → Nat) → Nat → Nat :=
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match x with
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| 0 => fun _ y => y
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| succ x => fun f y => f (y % succ x) (mod_lt _ (zero_lt_succ _)) (succ x)
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@[extern "lean_nat_gcd"]
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def gcd (a b : @& Nat) : Nat :=
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WellFounded.fix (measure id).wf gcdF a b
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@[simp] theorem gcd_zero_left (y : Nat) : gcd 0 y = y :=
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rfl
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theorem gcd_succ (x y : Nat) : gcd (succ x) y = gcd (y % succ x) (succ x) :=
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rfl
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@[simp] theorem gcd_one_left (n : Nat) : gcd 1 n = 1 := by
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rw [gcd_succ, mod_one]
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rfl
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@[simp] theorem gcd_zero_right (n : Nat) : gcd n 0 = n := by
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cases n <;> simp [gcd_succ]
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@[simp] theorem gcd_self (n : Nat) : gcd n n = n := by
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cases n <;> simp [gcd_succ]
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end Nat
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